/*
 * zzllrr Mather
 * zzllrr@gmail
 * Released under MIT License
 */

wiki['Sequence Sum']=

detail('求和公式',Table([ZLR('前n项求和 记法 结果')],[
	[['a+(a+d)+(a+2d)','+⋯+(a+(n−1)d)','等差数列'].join(kbr),
		sum('i',1,'n','(a+(i-1)d)','',''),
		Eq([frac('(a + a_n)n',2,''),'na + '+frac('n(n-1)',2,'')+'d'])+kbr+'数列逆序，分别相加'
	],
	[ZLR3('a',ZLR(' q q ⋯ q'),ZLR('  ^2  ^{n-1}'),'+')+kbr+'等比数列 q≠1',
		sum('i',1,'n','aq^{i-1}','',''),
		Eq([frac('a - a_nq','1-q',''),frac('a(1-q^n)','1-q','')])+kbr+'数列×q，错位相减'
	],
	[zlr('','1 2 3 ⋯ n','+'),sum('i',1,'n','i','',''),frac(1,2,'')+'n(n+1)'],
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^2','+'),
		sum('i',1,'n','i^2','',''),
		Eq([frac(1,6,'')+'n(n+1)(2n+1)',
		frac(1,3,'')+'n(n+1)(n+'+frac(1,2,'')+')'])
	],
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^3','+'),
		sum('i',1,'n','i^3','',''),
		frac(1,4,'')+'n^2(n+1)^2'
	],
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^4','+'),
		sum('i',1,'n','i^4','',''),
		[frac(1,30,'')+'n(n+1)(2n+1)','(3n^2+3n-1)',
		'='+frac(1,5,'')+'n(n+1)(n+'+frac(1,2,'')+')','(n^2+n-'+frac(1,3,'')+')'].join(kbr)
	],
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^5','+'),
		sum('i',1,'n','i^5','',''),
		Eq([frac(1,12,'')+'n^2(n+1)^2(2n^2+2n-1)',
		frac(1,6,'')+'n^2(n+1)^2(n^2+n-'+frac(1,2,'')+')'])
	],
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^6','+'),
		sum('i',1,'n','i^6','',''),
		[frac(1,42,'')+'n(n+1)(2n+1)','(3n^4+6n^3-3n+1)',
		'='+frac(1,7,'')+'n(n+1)(n+'+frac(1,2,'')+')','(n^4+2n^3-n+'+frac(1,3,'')+')'].join(kbr)
	],
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^7','+'),
		sum('i',1,'n','i^7','',''),
		[frac(1,24,'')+'n^2(n+1)^2','(3n^4+6n^3-n^2-4n+2)',
		'='+frac(1,8,'')+'n^2(n+1)^2','(n^4+2n^3-'+frac(1,3,'')+'n^2-'+frac(4,3,'')+'n+'+frac(2,3,'')+')'].join(kbr)
	],
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^8','+'),
		sum('i',1,'n','i^8','',''),
		[frac(1,90,'')+'n(n+1)(2n+1)','(5n^6+15n^5+5n^4-15n^3-n^2+9n-3)',
		'='+frac(1,9,'')+'n(n+1)(n+'+frac(1,2,'')+')','(n^6+3n^5+n^4-3n^3-'+frac(1,5,'')+'n^2+'+frac(9,5,'')+'n-'+frac(3,5,'')+')'].join(kbr)
	],
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^9','+'),
		sum('i',1,'n','i^9','',''),
		[frac(1,20,'')+'n^2(n+1)^2(n^2+n-1)','(2n^4+4n^3-n^2-3n+3)',
		'='+frac(1,10,'')+'n^2(n+1)^2(n^2+n-1)','(n^4+2n^3-'+frac(1,2,'')+'n^2-'+frac(3,2,'')+'n+'+frac(3,2,'')+')'].join(kbr)
	],
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^{10}','+'),
		sum('i',1,'n','i^{10}','',''),
		[frac(1,66,'')+'n(n+1)(2n+1)(n^2+n-1)','(3n^6+9n^5+2n^4-11n^3+3n^2+10n-5)',
		'='+frac(1,11,'')+'n(n+1)(n+'+frac(1,2,'')+')(n^2+n-1)','(n^6+3n^5+'+frac(2,3,'')+'n^4-'+frac(11,3,'')+'n^3+n^2+'+frac(10,3,'')+'n-'+frac(5,3,'')+')'].join(kbr)
	],
	
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^m','+')+kbr+'等幂和',
		sum('i',1,'n','i^m','',''),
		Eq([frac(1,'m+1','')+'n^2(n+1)^2(2n^2+2n-1)',frac(1,6,'')+'n^2(n+1)^2(n^2+n-'+frac(1,2,'')+')'])
	],

	[zlr('','2 4 6 ⋯ 2n','+'),
	sum('i',1,'n','2i','','')+'\\\\=2'+sum('i',1,'n','i','',''),
	'n(n+1)'
	],

	[zlr('','1 3 5 \\\\⋯ (2n-1)','+'),
		[sum('i',1,'n','(2i-1)','',''),
		sum('i',1,'2n','i','','')+'-'+sum('i',1,'n','2i','',''),
		2+sum('i',1,'n','i','','')+'-'+sum('i',1,'n','1','',''),
		].join(kbr+'='),
		['n^2','方法1: 2n项之和减去n项和','方法2: 拆开通项多项式（下同）'].join(kbr)
	],

	[ZLR3('',ZLR('1 3 5 ⋯ (2n-1)'),'^2','+').replace('⋯',kbr+'⋯'),
		Eq([sum('i',1,'n','(2i-1)^2','',''),
		sum('i',1,'2n','i^2','','')+'-'+sum('i',1,'n','(2i)^2','','')]),
		
		frac(1,3,'')+'n(2n-1)(2n+1)\\\\='+frac(1,3,'')+'n(4n^2-1)'
	],
	[ZLR3('',ZLR('1 3 5 ⋯ (2n-1)'),'^3','+').replace('⋯',kbr+'⋯'),
		sum('i',1,'n','(2i-1)^3','',''),
		'n^2(2n^2-1)'
	],
	[ZLR3('',ZLR('1 3 5 ⋯ (2n-1)'),'^4','+').replace('⋯',kbr+'⋯'),
		sum('i',1,'n','(2i-1)^4','',''),
		''
	],
	[ZLR3('',ZLR('1 3 5 ⋯ (2n-1)'),'^5','+').replace('⋯',kbr+'⋯'),
		sum('i',1,'n','(2i-1)^5','',''),
		''
	],
	[ZLR3('',ZLR('1 3 5 ⋯ (2n-1)'),'^m','+').replace('⋯',kbr+'⋯')+kbr+'奇数项等幂和',
		sum('i',1,'n','(2i-1)^m','',''),
		''
	],
	['1⋅2+2⋅3+3⋅4\\\\+⋯+n(n+1)',
		sum('i',1,'n','i(i+1)','',''),
		frac(1,3,'')+'n(n+1)(n+2)'
	],
	['1⋅2⋅3+2⋅3⋅4+3⋅4⋅5\\\\+⋯+n(n+1)(n+2)',
		sum('i',1,'n','i(i+1)(i+2)','',''),
		frac(1,4,'')+'n(n+1)(n+2)(n+3)'
	],
	['1⋅2⋅3⋅4+2⋅3⋅4⋅5\\\\+⋯+n(n+1)(n+2)(n+3)',
		sum('i',1,'n','i(i+1)(i+2)(i+3)','',''),
		frac(1,5,'')+'n(n+1)(n+2)(n+3)(n+4)'
	],
	[['1⋅2⋅3⋯k+','2⋅3⋅4⋯(k+1)+','⋯+','n(n+1)(n+2)⋯(n+k-1)','连乘和'].join(kbr),
		[sum('i',1,'n','i(i+1)⋯(i+k-1)','',''),'='+sum('i',1,'n',prod('j',1,'k','(i+j-1)','',''),'','')].join(kbr),
		
		[frac(1,'k+1','')+'n(n+1)(n+2)','(n+3)⋯(n+k)',
			'='+frac(1,'k+1','')+prod('i',1,'k+1','(n+i-1)','',''),
			'='+frac(1,'k+1','')+frac('(n+k)!','(n-1)!',''),
			'='+frac(1,'k+1','')+'A^{k+1}_{n+k}',
		].join(kbr)
	],

	['1⋅2^2+2⋅3^2+3⋅4^2\\\\+⋯+n(n+1)^2',
		sum('i',1,'n','i(i+1)^2','',''),
		frac(1,12,'')+'n(n+1)(n+2)(3n+5)'
	],

	['1⋅2^2⋅3+2⋅3^2⋅4+3⋅4^2⋅5\\\\+⋯+n(n+1)^2(n+2)',
		sum('i',1,'n','i(i+1)^2(i+2)','',''),
		frac(1,10,'')+'n(n+1)(n+2)(n+3)(2n+3)'
	],


	[['1(n^2-1^2)+','2(n^2-2^2)+','3(n^2-3^2)+','⋯+','(n-1)(n^2-(n-1)^2)'].join(kbr),
		sum('i',1,'n-1','i(n^2-i^2)','',''),
		frac(1,4,'')+'n^2(n^2-1)'
	],


	[['2⋅1⋅2+4⋅2⋅3+8⋅3⋅4','⋯+2^n⋅n(n+1)'].join(kbr),
		sum('i',1,'n','2^i⋅i(i+1)','',''),
		'2^{n+1}(n^2-n+2)-4'
	],


	[['1-2+3-⋯','+(-1)^{n-1}n'].join(kbr),
		sum('i',1,'n','(-1)^{i-1}⋅i','',''),
		[piece([[frac(1,2,'')+'(n+1)','n为奇数'],['-'+frac('n',2,''),'n为偶数']]),
			frac('1+(-1)^{n-1}(2n+1)',4,''),
			'(-1)^{n-1}'+frac(1,2,'')+'n+'+kbr+'\\left(0(偶)或'+frac(1,2,'')+'(奇)\\right)',
		].join(kbr+'=')+
		
		kbr+['利用F(n)='+piece(['f(n) & n为奇数','g(n) & n为偶数']),
			frac('f+g',2,'')+'-(\\cos nπ)'+frac('f-g',2,''),
			frac('f+g',2,'')+'-(-1)^n'+frac('f-g',2,''),
			frac('f+g',2,'')+'+(-1)^{n-1}'+frac('f-g',2,''),
		].join(kbr+'=')
	],


	[['1^2-2^2+3^2-⋯','+(-1)^{n-1}n^2'].join(kbr),
		sum('i',1,'n','(-1)^{i-1}⋅i^2','',''),
		[
		'(-1)^{n-1}'+frac(1,2,'')+'n(n+1)',
		].join(kbr+'=')
	],

	[['1^3-2^3+3^3-⋯','+(-1)^{n-1}n^3'].join(kbr),
		sum('i',1,'n','(-1)^{i-1}⋅i^3','',''),
		[
		frac('-1+(-1)^{n-1}(4n^3+6n^2-1)',8,''),
		'(-1)^{n-1}'+frac(1,2,'')+'n^2(n+3\\/2)-'+kbr+'\\left(0(偶)或'+frac(1,4,'')+'(奇)\\right)',

		].join(kbr+'=')
	],

	[['1^4-2^4+3^4-⋯','+(-1)^{n-1}n^4'].join(kbr),
		sum('i',1,'n','(-1)^{i-1}⋅i^4','',''),
		[
		'(-1)^{n-1}'+frac(1,2,'')+'n(n+1)(n^2+n-1)',
		].join(kbr+'=')
	],

	[['1^m-2^m+3^m-⋯','+(-1)^{n-1}n^m','交错等幂和'].join(kbr),
		sum('i',1,'n','(-1)^{i-1}⋅i^m','',''),
		[
		'(-1)^{n-1}'+frac(1,2,'')+'n(n+1)(????)+',
			'\\left(0(偶)或(-1)^{(m-1)/2}'+frac(1,'2^{m-1}','')+'(奇)\\right)',
		].join(kbr)
	],

]


	
,'wiki').replace(/\n/g,br))+

detail(i18('Reference'),ol([
	'《数学手册》，高等教育出版社，1979.5 第1版',
	'《实用数学手册》，科学出版社，2006.1 第2版',
	'《数学指南——实用数学手册》，科学出版社，2012.1 第1版',
]),1)



